"Symplectic birational transformations of a plane (following J. Blank's paper and Usnich's papers)

We will talk about the group of transformations of a projective plane that preserves differential form $\frac{dx\wedge dy}{xyz}$ (symplectic group $Symp$). We will see that it is generated by $SL(2,Z)$, torus, and a special element $P$ of order 5. We will observe a subgroup of $Symp$ generated by $SL(2,Z)$ and $P$ and show one presentation of it. Also we will try to look at combinatorics related to $Symp$.